5 PRODUCTS OF UNIFORMIZABLE SPACES 17 We have pra(A) <= pra(A) = Aa for all a e I ((12.5.2) and (3.11.4)), hence A a Y[ Aa. Conversely, if a = (ax) e f| Aa, and if we consider an ele- a eI ae I mentary set U = J| Ua x fj Ea containing a, its intersection with A is aeH ael-H n (Aanujx n A.. aeH ael-H Since none of the Aa is empty, no Aa is empty and therefore U n A ^ 0, which shows that ae A. (12.5.5) Let z \-+f(z) = (/a(r))a e { be a mapping of a topological space F into E. For f to be continuous at a point ZQ e F, it is necessary and sufficient that each fa should be continuous at z0. & For any elementary set U = f] Ua x Yl Ea, we have aeH ael-H (12.5.6) Let (jc^n)) be a sequence of points of E. For a = (0a) to be a limit of the sequence (x(w)), it is necessary and sufficient that aa should be a limit of the sequence (prax^)n^i9 for each a e I. This is an immediate consequence of (12.5.5) and the definition of a limit. In particular, if all the Ea are equal to the same space F, then to say that a mapping u e F1 of I into F is a limit of a sequence of mappings (un)n^l of I into F, with respect to the product topology, means that for each a e I the sequence (t/n(a)) has u(oc) as a limit in F. In this situation we say that the map- ping u is a simple limit of the sequence (un), or that the latter converges simply to u. (12.5.7) If each space Ea is Hausdorff, then so is the product space E. For if x 7* y, there exists an index a e I such that pra^c ^ pra<y; hence, using (12.4.4), there exists X e La such that 4t, *(px<xx> PrajO ^ 0, and there- fore e^ A(JC, y) 7* 0. The result now follows from (12.4.4). (12.5.8) 77?e product E of a denumerable jfam/Vy ofmetrizable (resp. separable metrizable) spaces is metrizable (resp. separable and metrizable). First of all, E is Hausdorff by (12.5.7). To show that E is metrizable, apply (12.4.6) and the definition of pseudo-distances on a product. Now oo suppose that E = f] Ew, where the En are separable and metrizable. Then each w=lpen in E.